Received 26 March 2006; received in revised form 19 July 2007; accepted 20 July 2007 Available online 2 August 2007

Abstract An algorithm for the distribution of fresh vegetables in which the perishability represents a critical factor was developed. This particular problem was formulated as a vehicle routing problem with time windows and time-dependent travel-times (VRPTWTD) where the travel-times between two locations depends on both the distance and on the time of the day. The model considers the impact of the perishability as part of the overall distribution costs and a heuristic approach, based on the tabu search is used to solve the problem. The performance of the algorithm was veriﬁed using modiﬁed Solomon’s problems. Using parameters typical of the Slovenian food market, diﬀerent schedules were achieved, giving improvements of up to 47% reduction in perished goods. Ó 2007 Elsevier Ltd. All rights reserved.
Keywords: Vehicle routing problem; Distribution; Perishable food; Time-dependent; Loss of quality

1. Introduction The diﬃculty in preserving the nutritional characteristics of fresh food-stuﬀs during transportation presents a direct problem to food distributors where the perishability of the produce requires it to be handled in ways not necessarily conducive to the traditional view of cost eﬀective distribution activities. Fresh vegetables provide a representative example of perishable goods; the nutrition value and taste are at their best directly after harvesting, decreasing as time elapses until the food is spoilt. In the Slovenian market, a quick distribution of fresh vegetables is required since, of the approximately 151,571,000 kg of fresh vegetables consumed per year, between 50% and 70% are imported, mainly from Italy and Spain. When they arrive in Slovenia, they are ﬁrst delivered to the distribution centers and then distributed to the ﬁnal consumers who can acquire the product in opti*

mal conditions only 48 h after the harvesting. In the transportation of vegetables insulated vehicles are used for international transportation and non-insulated vehicles for the ﬁnal distribution. A detailed description of the vegetables consumed, the typical quantities and the distribuˇ tion process are presented in Vadnal and Bratusa (2002), Krizaj (2002) and Knavs (2003). ˇ To measure the decrease in the value of a load of fresh vegetables, we deﬁne the ‘‘quality” of the load. The quality is 100% when the load can be sold entirely at the current market price and the quality is 0% when the load has lost its commercial value. The loss of quality in the transportation of the vegetables represents a signiﬁcant cost for the companies. By Slovenian law, the recognized value of the loss is 8%, but occasionally the loss can be higher and have been reported by transportation companies as up to 23% (Krizaj, 2002). ˇ Goods-distribution companies try to reduce basic distribution costs based on the number of vehicles used, total distance-traveled and the total travel-time. The standard view is that in addressing basic distribution issues distributors also

Notations Ni Cji number of the customers for vehicle i load transported by the vehicle i on the path between the customer j and the customer j + 1 Travel-timej(j+1) time spent by vehicle i between the start of servicing of customer j and the start of the servicing of customer (j + 1) Sk arrival time of vehicle k at customer i i DelayedCostFactori cost for a minute of lateness deﬁned for customer i NV number of vehicles Distancei distance in km traveled by the vehicle i Timei time in minutes that the vehicle i spend on the route Penaltyi time of delay in minutes that the vehicle i accumulate on the route and calculated as in (2) LoadÂTimei loss of quality at vehicle i, calculated as in (1) k1, k2, k3 and k4 parameters that represent the weight of each part of the total cost.

indirectly reduce the loss of quality. In this paper, we present a model for the representation of the loss of quality and consider it as part of the overall distribution costs. We model the distribution problem between the distribution centers and the customers (retailers) as a vehicle routing problem with time windows and with time-dependent travel-times (VRPTWTD) where the travel-times between two locations depends on both the distance and on the time of day. To minimize the overall distribution cost, the objective function must not only model the number of vehicles, the total distance-traveled and the total travel-time, but additionally the loss of quality of the load. 2. Literature review Diﬀerent methods have been developed to solve the vehicle routing problem (VRP) and vehicle routing problem with time windows (VRPTW). The routes should be chosen to minimize the total distribution cost. A detailed description of these and other related problems including a literature review of the methods are provided in Toth and Vigo (2002). In the VRPTW we know the location, the demand and the delivery window (the earliest and latest time that the customer is able to accept the load) of each customer. The solution is presented by one or more routes, each associated to one vehicle (a route begins at the depot, traverses a subset of customers in a speciﬁed sequence and returns to the depot). Each customer must be assigned only to one vehicle and the load must not exceed the vehicle capacity. Various works address the transportation of foodstuﬀs, dealing with pertinent issues to our problem. Tarantilis and Kiranoudis (2001) analyzed the distribution of the fresh milk. They formulated the problem as a heterogenous ﬁxed ﬂeet vehicle routing problem; this is a VRP with vehicles that have diﬀerent capacity. A threshold-accepting based algorithm was developed aiming to satisfy the distribution needs of the company, allowing them to schedule their distribution many times a week. Tarantilis and Kiranoudis (2002) presented a real-life dis-

tribution problem of fresh meat in an area of the city of Athens. They formulated the problem as an open multidepot vehicle routing problem. They presented a new stochastic search meta-heuristic algorithm belonging to the class of threshold-accepting algorithms. Hwang (1999) presented an eﬀective distribution model for determining optimal patterns of food supply and inventory allocation for famine relief areas. He modeled a VRP that incorporated inventory allocation and optimal distribution based on minimizing the deprivation and starving instead of travel distance or time. Prindezis, Kiranoudis, and MarinosKouris (2003) presented an application service provider, to be used for central food markets, which coordinates and disseminates tasks and related information for solving the VRP. For the solution of the VRP they used a metaheuristic technique based on the tabu search. They tailored their software to the road network of Athens and applied it to the integrated-logistics problem of deliveries to the 690 retail companies that comprise the Athens Central Food Market. They used a two-phase algorithm to solve the VRP. In the ﬁrst phase, a route construction algorithm was used and in the second phase a tabu search was used to improve the given solution. Faulin (2003) presented the implementation of the mixed algorithm procedure that uses heuristic and exact subroutines in the solution of a VRP having speciﬁc constraints related to companies in the agribusiness ﬁeld. In all the above models, the travel-time between the customers was constant and none of them took into account the speciﬁc degradation of quality during transport. Amponsah and Salhi (2004) presented an eﬃcient heuristic for the routing problem of the collection of garbage. They considered it as a capacitated arc routing problem (CARP) and minimize the environmental eﬀect due to the smell of the garbage and total collection cost in a bi-objective model. In CARP the customer (garbage to be collected) is associated to the arcs and not to the nodes as in the VRP. The smell of the garbage gets worse with time and so it is necessary to remove it as early as possible,

A. Osvald, L.Z. Stirn / Journal of Food Engineering 85 (2008) 285–295

287

which is analogous to food deterioration. They deﬁned inconvenience due to smell by the total quantity of garbage over time. They used a look-ahead strategy that gives the possibility to choose from a pool of solutions the one which best solves the problem. In most real-world distribution problems, it is important to consider the ﬂuctuations of the travel-time in the solution of the problem. Travel-time plays an important role in the distribution of the perishable goods, since its ﬂuctuations may extend the time that the goods spend on the vehicles. Diﬀerent representations of the ﬂuctuations of the travel-times between the customers have been reported and diﬀerent extensions of VRP have been proposed to address the ﬂuctuations in travel-times. Routing problems with stochastic travel-times are presented by Gendreau, Laporte, and Seguin (1996), Fu (2002), Hadjiconstantinou and Roberts (1998) and Kenyon and Morton (2003). The routing problem that includes time of day-dependent travel-time was ﬁrst introduced by Malandraki and Daskin (1992). They considered the time-dependent traveling salesman problem (TSP), that is a simpliﬁed VRP problem in which all customers must be visited by only one vehicle. Bentner, Bauer, Obermair, and Morgenstern (2001) presented a TSP in which they have considered a zone in the city center with traﬃc jams in the afternoon and show how simulated annealing and threshold-accepting algorithms are able to handle such time-dependent problems. Park (2000) presented the time-dependent VRP in which the travel speed between two locations depends on the route and the time of the day. They proposed a model for estimating the time varying travel speed. Ichoua, Gendreau, and Potvin (2003) proposed a time-dependent model for the VRPTW. The model that they developed is based on time-dependent travel speeds and satisﬁes the ﬁrst-in-ﬁrst-out (FIFO) property. They extended the tabu search heuristic to solve the problem and showed that the

time-dependent model provides substantial improvements over a model based on ﬁxed travel-times.

3. Quantiﬁcation of quality loss It is diﬃcult to deﬁne the quality models for perishable foods since food degradation depends on various parameters and there exist many criteria on how to measure it. Sloof, Tijskens, and Wilkinson (1996) presented a study on how to estimate the quality of the perishable products and use the eﬀect-decomposition in the presented models. ˇ Verbic (2004) presented a study of the parameters aﬀecting preservation of perishable goods in the cold chain. The studied parameters are the intensity of deterioration of goods at given levels of production. Bogataj, Bogataj, and Vodopivec (2005) studied the eﬀects of perturbations in a supply chain (production or distribution part) on the stability of perishable goods in order to keep the product at the required level of quality and quantity for the ﬁnal delivery. They present a model in the time domain and compare it with the formulation in the frequency space. The above methods are very complex and require the estimation of many parameters which are not available in the distribution process. Therefore, we have extended the simple linear model proposed by Pawsey (1995) to estimate the decrease of fresh vegetables quality. We assume that any perishable product has a limited life-span under given conditions which is divisible into two stages. The point t = 0 represents the optimal condition of the vegetables; generally at the moment of harvest. During the ﬁrst period of apparent quality stability (from 0 to A in Fig. 1), the quality is reducing but there are no discernible changes. At point A noticeable changes start in one or more of the quality parameters. During the second stage (from A to B in Fig. 1), the changes continue and at point B the product is unacceptable. In real-life the distribution is sel-

dom completed in the 0–A period, and it is normal that the quality of the goods is reduced before delivery. The time that elapses between harvesting and distribution is often quite high. In Slovenia, it is usually greater than 24 h, especially for imported vegetables. Most vegetables begin to visibly degrade after this period. Considering these data, we have made the assumption that distribution in Slovenia starts at point A. Therefore the quality, Q [0,1], in the period A–B can be considered as a linear function as follows Q¼1À tÀA : BÀA

We also assume that Q / Saleable Quantity / Commercial Value: A loss of quality of 20% can be related to either a load where 20% of transported quantity is completely damaged and 80% is in perfect conditions, a load where the whole transported quantity was evenly damaged so that it could be sold only at 80% of its original price, or a combination of these. Slovenian distribution companies and wholesalers estimate that the loss of quality of fresh vegetables during the transportation is on average between 8% and 23%. Based on these estimates we have assumed that the distribution schedule organized without considering the loss of quality in the objective function always ends before point B is reached. It is diﬃcult to deﬁne the actual rate of deterioration of the quality of a load over time, as it is dependent on the storage conditions and type of vegetable transported. However, it is not necessary to estimate the rate of deterioration to compare two diﬀerent schedules of the same load. If we know the ﬁnal quality of one of the schedules then we can estimate the relative ﬁnal quality of the others. To compare the loss of the quality of two diﬀerent schedules we use the parameter LoadÂTimei Load Â Timei ¼
N ¼iÀ1 X j¼0

to the absolute loss in kg of saleable quantity. This means for example, that an increase of 20% of LoadÂTime causes a 20% reduction of saleable quantity. If we have two diﬀerent schedules and know the loss of the quality of one, then we can estimate the eﬀective loss of the second schedule using this parameter. The loss of quality represents an additional cost for the sellers (the complete chain of wholesalers, middlemen and shopkeepers in the distribution process) because a part of the purchased goods cannot be sold on the market or it can be sold, but not with the best market price. The acquisition and the selling price of vegetables may vary during the year dependent on the vegetable type. We have calculated an average representative price for vegetables considering the typical quantities consumed of the ˇ diﬀerent vegetables available (Vadnal & Bratusa, 2002) and considering the average prices on the Slovenian market (selling and acquisition price) over the period May–Octoˇ ber 2005, reported in the Slovenian magazine Kmecki glas that weekly publishes the prices of the more important vegetables on the market. On the basis of these average prices we have calculated the ‘‘sell price” of the vegetables reported in Table 1. 4. Mathematical model for the transportation of the vegetables We model the problem of the distribution of the vegetables between the central depot and the ﬁnal customers (retailers) as a VRPTWTD. In our model we assume: homogeneous ﬂeet of vehicles with a limited capacity; loss of the load quality during delivery as a linear function of time; time-dependent travel-times between the customers; soft time windows for all customers (time window may be violated at a penalty cost); hard time windows for the depot (violation of time window is not allowed). In the transportation of vegetables in Slovenia, the time windows are usually only indicative. This and the fact that the travel-times ﬂuctuate were the reasons to consider soft time windows. If [ai, bi] represents the time window of customer i then, if the vehicle arrives at customer i before time ai, it must wait up to time ai before starting its service. If it arrives after bi, a lateness penalty is incurred and is calculated using

TravelTimejðjþ1Þ Á C ji ;

ð1Þ

where Ni, number of the customers for vehicle i; Cji, load transported by the vehicle i on the path between the customer j and the customer j + 1 (kg) and TravelTimej(j+1), time spent by vehicle i between the start of servicing of customer j and the start of the servicing of customer (j + 1) (min) Cij can be calculated using C ji ¼
Ni X k¼j

ck ;

where ck is the load to be delivered to customer k (kg). LoadÂTimei represents the quantity in kg delivered by the vehicle i to its customers, multiplied by the time in minutes needed for the delivery. When the linear dependency of quality versus time is ﬁxed, this quantity is proportional

A. Osvald, L.Z. Stirn / Journal of Food Engineering 85 (2008) 285–295

289

( Penaltyi ¼

ðS k À bi Þ Á Delayed Cost Factori i 0

S k P bi i S k 6 bi i

ð2Þ arrival time of vehicle k at customer i (min) and where DelayedCostFactori, cost for a minute of lateness deﬁned for customer i (€/min). The objective function to minimize total cost is NV NV X X TotalCost ¼ k 1 Distancei þ k 2 Timei Sk , i
i¼1 i¼1

þ k3

NV X i¼1

Penaltyi þ k 4

NV X i¼1

Load Â Timei ;

ð3Þ

where NV, number of vehicles; Distancei, distance in km traveled by the vehicle i; Timei, time in minutes that the vehicle i spend on the route; Penaltyi, time of delay in minutes that the vehicle i accumulate on the route; LoadÂTimei, loss of quality at vehicle i, calculated as in (1) and k1, k2, k3 and k4, parameters that represent the weight of each part of the total cost. In the minimization of the objective function, it is not the value given to each k that is important, but the ratio between them. In the preliminary tests we have found that the most representative ratio between k1, k2 and k3 for our problem is equal to 1. The units of the parameters are as follows: k1 [€/km], k2 [€/min], k3 [–], k4 [€/ (min kg)]. By reducing the total LoadÂTimei we reduce the time that the vegetables spend on the vehicles. This quantity does not take into account the time-traveled with empty vehicles; therefore, it is not the same as the Timei quantity. The travel-time between the customers is time-dependent, i.e., the travel-time between two locations depends on both the distance and on the time of day. Due to the time-dependency of the travel-times, the distance-traveled is also time-dependent (diﬀerent routes for diﬀerent times can be chosen in order to reduce the travel-time). To simplify the problem, we consider only four diﬀerent time periods and suppose that the travel-time and the distance-traveled between the customers in each time period is constant (it is a step function of time). To calculate the distance-traveled and the travel-times between the customers in each time period, we use the position of the customers. If the customers are deﬁned on the real road network, we use the network characteristics and solve the time-dependent shortest path problem as in our previous work (Osvald, 2005). Otherwise, we use the corrected Euclidean distances. To assure that the ﬁnal solution of the VRPTWTD satisﬁes the FIFO property deﬁned in Sung, Bell, Seong, and Park (2000) it is necessary to have continuous traveltimes. Therefore, we round oﬀ travel-times in order to achieve time-continuous travel-times. An example of rounding of travel-times where four time periods during the day are used is presented in Fig. 2. If a vehicle starts

Fig. 2. Rounding of the travel-times for the four time periods used in the solution.

to travel on one arc near the end of one time period, then it cannot go the whole length of the arc before the end of the time period. The vehicle will therefore use two diﬀerent speeds to cover the complete distance of the arc. If TimeA is the time necessary to go the whole length of the arc in the time period [T0, T1] and TimeB is the time necessary to go the whole length of the arc in the time period [T1, T2], then the rounding of the travel-time in the ﬁrst and second time period is represented as follows
8 > TimeA > > > > TimeA þ > < t 6 T 1 À TimeA ; T 1 À TimeA < t 6 T 1 ; T 1 < t 6 T 2 À TimeB :

TravelTime ðtÞ ¼

TimeB ÀTimeA Á > TimeA > > ðt À T 1 þ TimeA Þ > > > : TimeB

Similar representations can be deﬁned for other time periods. 5. Solution of the problem To solve the problem, we have extended our previous method in Osvald (2005), developed to solve the VRPTWTD. The outline of the method used is

290

A. Osvald, L.Z. Stirn / Journal of Food Engineering 85 (2008) 285–295

Repeat for different Start solutions (different Start Customers and different insertion costs) Find a feasible route: Solve classical VRPTW with an average travel-time and travel distance Calculate the effective arrival time using time-dependent travel-times For each not feasible route Repeat until route is not feasible Remove customer from the route End Repeat End For Insert the removed customers in other routes (if it is not possible, initialize new routes for the removed customers) end Repeat Select the three best routes For each route Route reduction (reduce the number of routes) Insert stopover at the depot into each route where possible Insert free vehicles into the route Improve the solution with tabu search end For

Initially a sequential constructive heuristic that solves a classical VRPTW is used; this approximates the traveltime and the travel distance between each pair of customers with the average travel-time and the average travel distance. Diﬀerent initial solutions are calculated, since Braysy (2001) showed that changing the initial customers ¨ and using diﬀerent functions for the evaluation of the insertion costs of the customers (insertion cost represents the additional cost of the route when a new customers is inserted into it) yields much better solutions. The insertion cost is represented by a linear function which includes additional time and delay of the route, when a new customer is inserted. At each iteration (i.e., in each calculation of the initial solution), diﬀerent coeﬃcients for the additional time and additional delay are considered. Then a correction procedure is used to take into account the actual time dependencies of the travel-times. In this procedure, the customers’ delivery times are recalculated using time-dependent travel-times. When necessary, some customers are relocated in the route, or removed from it in order to make the route feasible. The removed customers are, when possible, reinserted in other routes. Otherwise, a new route or routes are initialized for the removed customers. In the generation of the initial solutions we do not consider the loss of quality (k4 = 0), since preliminary tests have shown that the consideration of these losses gives initial solutions with high lateness and high distances which cannot be always reduced in the following steps of the used method. After the generation of the initial solution we consider the objective function, including the loss of quality (k4 6¼ 0).

To further reduce the loss of saleable quantity it is necessary to reduce as far as possible the time that the vegetables spend on the vehicles. This could be achieved if vehicles perform shorter routes and return more often to the depot to take appropriately preserved vegetables. In this case the total traveled time would be higher, but the time that the vegetables spend on the vehicles is reduced. So, in the second part we perturbate the solution. First, we force the vehicles to go back to the depot in order to take appropriately preserved vegetables (at the depot, the vehicle has to wait for a speciﬁed time to complete loading operations). Therefore we introduce, where possible, new stopovers at the depot during the route with a deﬁned stop time at the depot. The outline of the procedure is Current location = last location Repeat until Current location = 0 Insert a ‘‘virtual’’ depot into the route at current location If the depot time window is not violated Current solution = new solution Current location = Current location-1 End repeat Second, since it is not always possible to add new stopovers at the depot in the existing routes (indicated as 00 in Fig. 3), we also add empty vehicles which have at least one additional stopover at the depot (only in this way we can be sure that the solution in which a vehicle stops at the depot, is taken into account in the optimization process). Usually we add two vehicles to each solution in order to extend the space of the possible solutions.

A. Osvald, L.Z. Stirn / Journal of Food Engineering 85 (2008) 285–295

291

Fig. 3. Example of insertion of new stopovers at the depot and new empty vehicles.

The next step consists of applying the improvement heuristic based on the tabu search, which is a local-search method that uses memory structures. This method starts from an initial solution and at each iteration generates a neighborhood, i.e., a set of solutions around the current solution. The best solution in this neighborhood becomes the next current solution, even if it is worse than the current one. By further investigating the worse solutions, it is possible to avoid the trap of local optimum solutions. To generate a neighborhood we use a cross-exchange move in which two segments from two routes are taken and then swapped. An example is presented in Fig. 4. In the VRPTWTD it is not always possible to achieve a feasible solution starting from the infeasible one. So we consider only feasible solutions in the neighborhood generation. The improvement heuristics is executed three times. The maximum number of iterations was ﬁxed at 2000. To develop the improvement heuristic based on the tabu search we have used EasyLocal++, an object-oriented framework for the design and analysis of local-search algorithms, as presented in Di Gaspero and Schaerf (2001). Osvald (2005) provides detailed presentation of the neighborhood generation (including the software code) and the calculations of other parameters (e.g., the length of the tabu list, maximum number of iterations). In the improvement heuristic most of the additional stopovers at the depot and the additional vehicles are subsequently removed in order to reduce the total distribution cost.

However, the remaining additional stopovers and vehicles generate a quality saving greater than their additional cost. 6. Performance of the solution method 6.1. Deﬁnition of the data set Ideally the method would be tested on real data, considering real customers and traﬃc conditions. Unfortunately, the wholesalers operating in Slovenia would not make such commercially sensitive data such as the eﬀective arrival time at the customers, the exact travel-time that the vehicles spend on the route and the exact delivered order quantity available in the public domain. We know the total distributed quantity of vegetables for different areas, but not individual customer orders. Additionally, we do not know the exact location of all customers and acquisition of this data would require a great eﬀort including the collection of all addresses and their mapping on a vector map. Therefore, we decided to use some representative data sets in which the distribution of the customers and requests are known. We chose the classical Solomon’s instances (Solomon, 1987). Each of the 56 instance has 100 customers generated within a 100 by 100 km square area. The distances between the customers are Euclidean. The problems are partitioned into six diﬀerent sets, namely C1, C2, R1, R2, RC1 and RC2. The customers are uniformly distributed in the problems of type R, clustered in groups in the problems of type C and mixed in the problems of type RC. In the problems of type 1, a small number of customers can be serviced on each route due to a narrow time window at the depot, as opposed to problems of type 2 where each route may have many customers. These data sets are suﬃcient to test the validity of the solutions since they are constructed so that they each consider a diﬀerent distribution of the customers. Also, the best solutions for number of vehicles used, total distance-traveled and the total travel-time are known for these problems so we can compare these reference case solutions to solutions if the consideration of load quality is added.

Fig. 4. Example of a cross-exchange move.

292

A. Osvald, L.Z. Stirn / Journal of Food Engineering 85 (2008) 285–295

In order to achieve representative data for the Slovenian market, we have analyzed various independent wholesalers that operate in Slovenia. The largest has three distribution centers and about 700 retailers dispersed across Slovenia. The other wholesalers are smaller and usually operate within a limited footprint, not covering the whole distribution area. The most common vehicle in use is not insulated, has a capacity of 10,000 kg, and the operating cost is about 0.6 €/km, so this was used in the data sets. 6.2. Deﬁnition of the tests To evaluate the performances of the proposed optimization method we performed the following tests: A1: Solution of the classical Solomon’s instances using the method presented by Osvald (2005) with k1 = k2 = k3 = 1 and k4 = 0 in Eq. (3) (i.e., we do not consider the loss of quality). A2: Solution of the classical Solomon’s instances using the method presented by Osvald (2005) with k1 = k2 = k3 = k4 = 1 in Eq. (3) (i.e., consideration of loss of quality). A3: Solution of the classical Solomon’s instances using the method presented in Section 5 with k1 = k2 = k3 = k4 = 1 in Eq. (3). A4–A6: Same as in A1–A3, respectively, but using modiﬁed Solomon’s instances considering time-dependent travel-times. To take into account the time-dependent travel-times, we consider four time periods and generate four diﬀerent matrices, multiplying the origin destination (OD) matrices in the Solomon instances with four diﬀerent correction factors (namely 0.8, 1.2, 0.85 and 1.15). Each matrix represents the travel-time in one of the four time periods. We deﬁne the time periods so that for each instance the average travel-times between the customers are as the same as those in the Solomon’s instance. Finally we rounded the traveltimes to give a continuous travel-time between the customers as shown in Fig. 2. In tests A3 and A6 we set stops at the depot to be three times the maximum time necessary to unload the order at AdditionalProfitAi ¼ ðLoss Reduction iAi Þ Á EffectiveLoss 100

NV is the number of vehicles used in test A1 and reported in Table 2 (e.g., 11.08 vehicles used for R1, 10 used vehicles for C1). TotWC is the average load weight of each vehicle. The weight capacity of the vehicles is usually not completely used due to restrictions such as volume of the load, the distribution of the customers across the delivery area and time windows. To quantify the eﬀective use of the capacity, we calculated WR ¼ TotalTransportedQuantity ; VehicleCapacity Á NV

where the TotalTransportedQuantity and the VehiclesCapacity is expressed in the units deﬁned in the Solomon’s instances. Further we restricted vehicle weight capacity to 80% of the nominal value. Therefore, the total weight capacity can be calculated as TotWC ¼ 0:8 Á WR Á 10; 000 kg: 7. Results and discussion Tables 2–7 present the results for tests A1–A6. In these tables, for each set of Solomon’s instances, we report: – Solomon’s class: name of the set of Solomon’s instances. The results reported in the tables are the average value of all instances of the set. – Number of vehicles: the average of the number of vehicles used for all instances of the set. – Distance: the average of the total distances traveled in km for all instances of the set. – Time: the average of the total traveled times in minutes for all instances of the set. – Lateness: the average of the total lateness in minutes for all instances of the set. – LoadÂTime: the average of the loss of quantity for all instances of the set calculated using Eq. (1). – Loss reduction: the percentage reduction of LoadÂTime in the tests where we consider loss of quality in the objective function compared to tests that did not consider loss of quality. – Additional proﬁt: the proﬁt that could be achieved, when due to the reduction of the losses, more vegetables can be sold to the customer; it is calculated using

Â TotLoad Á SellPrice þ ðTotalDistanceAi À TotalDistanceA1 Þ Á PriceKm the customers. This upper-limit was deﬁned, since no exact information about the real time required at the depot was available. The total transported quantity (TotLoad) for each set of problems was calculated as follows TotLoad ¼ TotWC Á NV: ð4Þ

ð5Þ

where EﬀectiveLoss, the percentage loss of saleable quantity in the schedules (A1 or A4) and TotLoad, the total transported quantity or the total demand in kg calculated using Eq. (4). SellPrice and PriceKm are deﬁned in Table 1. Table 3 shows the results for the test A2 in which we have considered the loss of quality. Here we have an

increase of vehicles used of up to 7.7%, an increase of distance-traveled between 13% and 48% and an average reduction of loss of quality equal to 17%. To quantify the improvement of the achieved solution, we have calculated the additional available proﬁts (additional proﬁt) considering two diﬀerent percentages of loss of saleable quantity in the schedule A1 (eﬀective loss = 8 and eﬀective loss = 23 which represent the loss recognized by law and the loss occasionally experienced). The calculated quantities are signiﬁcant; they show that the reduction of the loss of quality oﬀers signiﬁcantly higher savings than the increased cost due to the increased distance-traveled. Of course, in real-life this additional proﬁt cannot be always achieved because the demand on the market is limited, but our results clearly indicate the value that these losses of saleable quantities of vegetables represent on the market. Since this distribution activity is undertaken many times per weeks, the compound impact of these additional proﬁt achieved across the year could be signiﬁcant. One of the parameters that it is also necessary to take into account is the total lateness. Even though we have a signiﬁcant increase of the total lateness (from 0 to 4800 min for the problem C2), it remains acceptable in all instances except C2. In reality, a total lateness of 1000 min represents an average delay per customer of 10 min which is probably acceptable. Table 5 shows the results for test A4. The LoadÂTime represents the average of the loss of quantity for the scheduling achieved in test A1, considering time-dependent travel-times (we consider the same customer sequence as in test A1, but we use the time-dependent travel-times to calculate the arrival time at the customer). The reduction of the loss of quality that we achieve considering time-dependent travel-times is nearly 20%. In reality, the travel-time plays an important role in the distribution of the perishable goods, since its ﬂuctuations may extend the time that the goods remain on the vehicles and the resultant eﬀect on the total loss of quality of the load. By considering the loss of quality in the solution of the problem (test A4), we achieve a signiﬁcant reduction of the loss of quality as presented in Table 5. Tables 4 and 7 present the results for tests A3 and A6 in which the method presented in Section 5 was used. In test A3, the increase of vehicle numbers is more signiﬁcant than for A2 as is the improvement in the reduction in total losses. The additional proﬁt increased nearly two times in comparison to test A2 for the average results, which dem-

onstrates the advantages in the proposed method. The lateness is higher for classes C1 and C2 (not always acceptable in real-life) and acceptable in all other classes. In A6 (Table 7) we have an average improvement of 7% on the total loss of quality compared to test A5. Moreover, there is a reduction of vehicle numbers and a reduction of the distance-traveled. The introduction of new stopovers (presented in Section 5) allows, in some cases, the generation of a good starting solution for the tabu search which can generate a ﬁnal solution with a very low number of vehicles and a short distance traveled. The additional proﬁts are higher than those achieved in the time-independent test A3. In our method we have considered an upper-limit for the time at the depot. A reduction of the stop time at the depot improves the reduction of the loss of the quality, since more stopovers at the customers are possible and so the time that the vegetables pass on the vehicle is reduced. 8. Conclusions The eﬃciency of the distribution process of perishable foods, where time the food spends on the vehicles is the most signiﬁcant and crucial factor, represents an important problem in food distribution. For a distribution network based on the Slovenian vegetable market, for constant travel-times the proposed method gave solutions with 27.9% average extra reductions in the loss of quality relative to the solution based on the model in which the loss of quality was not taken into account. The savings increased on average to 40% when time-dependent travel-times were taken into account. References
Amponsah, S. K., & Salhi, S. (2004). The investigation of a class of capacitated arc routing problems: The collection of garbage in developing countries. Waste Management, 24, 711–721. Bentner, J., Bauer, G., Obermair, G. M., & Morgenstern, I. (2001). Optimization of the time-dependent traveling salesman problem with Monte Carlo methods. Physical Review E, 036701-1–036701-8. Bogataj, M., Bogataj, L., & Vodopivec, R. (2005). Stability of perishable goods in cold logistic chains. International Journal of Production Economics, 93–94(1), 345–356. Braysy, O. (2001), Eﬃcient local search algorithms for the vehicle routing ¨ problem with time windows. Working Paper, SINTEF Applied Mathematics, Department of Optimization. Di Gaspero, L., & Schaerf, A. (2001). Writing local search algorithms using EASYLOCAL++. In S. Voß & D. L. Woodruﬀ (Eds.).